Exercise 11, § 1

Let $a_i = (a_{ij})$ ($1 \leq i \leq m,\ 1 \leq j \leq n$) be $m$ points of $\mathbf{R}^n$ and let $q$ be an integer > o. For each point $k = (k_i)$ ($1 \leq i \leq m$) of $\mathbf{R}^m$ with integer coordinates, not all zero, satisfying the inequalities $0 \leq k_i \leq q$, consider the point $x_k = (x_{j,k})$ ($1 \leq j \leq n$) of $\mathbf{R}^n$, all of whose coordinates belong to $[0, 1[$ and which is congruent mod $\mathbf{Z}^n$ to $\sum_{i=1}^m k_i a_i$, i.e., the point with coordinates

If $p$ is the smallest integer such that $q + 1 \geq p^{n/m}$, show that there are two distinct points $k, k'$ such that $x_k$ and $x_{k'}$ both belong to the same cube of side $1/p$ (same method as Exercise 10). Deduce that there exist $m$ integers $p_i$, not all zero, and $n$ integers $r_j$ ($1 \leq j \leq n$) such that $0 \leq p_i \leq q$ for $1 \leq i \leq m$, and

Use this result to give another proof of Proposition 2 of no. 1 (the "pigeon-hole principle").